Joint share cap on the top k AD terms (cycle 84)

The winsorize sweep (stuff/topk-winsorize-sweep.md) ended by naming this as the next probe: instead of moving draws, cap the k largest AD gate terms *jointly* at half the total, leaving the draws, the standardization and the other n-k terms untouched. k = 0 is the uncapped baseline, in the same table.

Seed 84003, 20000 sims per n per law (normal and t(3)), n = 25, 50, 100, cap share C = 1/2. Power is P(stat > normal-data p95 of the same variant at the same n) on t(3), the cycle 83 convention used by the other two items. Script: stuff/topk-joint-share-cap.py (pure python, ~8 s); data: stuff/topk-joint-share-cap.csv.

Verdict (rule 2)

• n = 25: none. Best is k=3 at 1.04, still above 2/3; k=3 and k=5 also lose 0.022 power, outside 0.02.
• n = 50: none. Best is k=5 at 1.16 (power -0.017, inside 0.02).
• n = 100 (power only by the direction; var/mean^2 reported for context): none; k=5 gives 1.07.

No k reaches var/mean^2 at most 2/3 at any n. Unlike the winsorize sweep, the joint share cap keeps power: every row is within 0.02 of uncapped except k=3, 5 at n=25 (-0.022, about 1.5 SE of 0.0034 at 20000 sims).

Where the excess lives

• In the top 1% of t(3) sims the capped top k still hold 0.42 to 0.48 of the statistic, i.e. the cap binds in essentially every extreme sim. The rest (n-k terms) carries the other half, and it is not small: var/mean^2 sits near 1.1 to 1.2 for k >= 2 and moves little from k=2 to k=5. The residual excess is spread across the bulk of the terms, consistent with cycle 83's "remaining terms, not the largest".
• k=1 is not enough: with only one term capped a second extreme draw keeps the tail (3.6 at n=50, 6131 at n=100 from a handful of sims with two huge terms). k >= 2 removes that.

Caveats

• Capping is pointwise downward, so var/mean^2 can rise above uncapped (n=100, k=1) when the capped mean is much smaller than the outlier-driven uncapped mean; the n=100 uncapped value (26) is itself unstable at this sims count.
• One cap share (1/2) only; a lower C would bring the top k share down but cannot touch the n-k bulk, which already sits near var/mean^2 of 1 on its own, so lower C is unlikely to reach 2/3.

Residual by rank: after capping the top k, the excess sits in the cap-setting draw and the next few ranks, and no k meets both halves of the direction

**Answer to the cycle 84 direction (rule 2 verdict), independent replication at seed 84003.** No k satisfies both halves at either n. Smallest k with capped var/mean^2 at most 2/3 while normal-calibrated power at n=25, 50, 100 stays within 0.02 of uncapped:

• n=25: none. var/mean^2 falls 51.1, 1.20, 0.52, 0.35, 0.23 for k=0, 1, 2, 3, 5, so the moment bound is met from k=2, but k=1 already loses 0.170, 0.176, 0.092 of power at n=25, 50, 100.
• n=50: none. var/mean^2 falls 398, 16.7, 1.13, 0.65, 0.39; the moment bound is met from k=3, which loses 0.306, 0.412, 0.309 of power.

This matches member-2's seed 84002 run (none at both n, moment bound from k=2 at n=25 and k=3 at n=50) to within sampling noise: k=1 var/mean^2 1.20 vs 1.15 at n=25 and 16.7 vs 10.3 at n=50 (the n=50, k=1 variance is still driven by a few huge sims), power losses within 0.03 at every (k, n).

Where the residual lives. The second table splits the post-cap AD sum in the top 1% of t(3) sims by original |draw| rank. Uncapped (k=0) the largest draw carries 0.99 to 1.00 of the sum. After k=1 the flattened top draw still holds 0.33 to 0.40, and the rank-2 draw that sets the cap level holds another 0.31 at n=25 and 0.50 at n=50: the two together are 0.65 and 0.90. As k grows the cap-setting draw's share drops (0.14, 0.10, 0.04 at n=25 for k=2, 3, 5) and the mass spreads into ranks k+2..k+5 and the rest, which together reach 0.74 at n=25 and 0.61 at n=50 by k=5. So the residual excess is a moving target: whichever draw sets the cap inherits the dominance, and capping it in turn hands the excess down the order statistics. On normal data the same cap leaves the top k and cap-setting draw with only 0.05 to 0.25 of the top-1% sum, so the shares in the t(3) table are a t(3) feature, not an artifact of the cap.

Consequence for the gate: the t(3) signal the gate detects is the same extreme tail the winsorize removes, which is why power falls at roughly the rate var/mean^2 does. Any fix that reaches 2/3 at these n has to act on the ordered tail as a whole (a rank-weighted or trimmed-sum statistic), not on a fixed number of top draws.

Inputs (rule 1)

• Statistic, standardization, t(3) sampler and cap rule identical to stuff/topk-winsorize-sweep.py (cycle 71/75 lower-tail weighted AD gate, n^-1/2 * sum_i (F_n(i) - Phi(z_i))^2 / Phi(z_i); variant k winsorizes the k largest |draw| to the (k+1)th largest |draw|, sign kept; k=0 uncapped).
• Script stuff/topk-residual-by-rank.py, output stuff/topk-residual-by-rank.csv, runs in about 4 s in pure python.
• Seed 84003 (per-stream 8400300 + n + {0 t(3), 1 normal}); 4000 sims per n per law; n = 25, 50, 100.
• Share columns: post-cap AD terms of the named draws divided by the AD sum, averaged over the top 1% (40) of t(3) sims by statistic value. "top k" is the rule 1 share column.
• Power: critical value is the normal-data p95 of the same variant at the same n; power is the fraction of t(3) samples above it; "diff vs k=0" is the power change against the uncapped gate at the same n. The k=0 powers 0.356, 0.588, 0.842 match cycle 83's 0.372, 0.585, 0.834 and cycle 75's 0.36, 0.60, 0.84. Power columns are per n and so repeat across the two n-blocks of the first table; the var/mean^2, p99 and share columns are the per-(k, n) quantities.

Rule 1 and rule 2 table: t(3), seed 84003, 4000 sims per n per law; k = 0 is the uncapped baseline

Where the residual lives: post-cap AD share by original |draw| rank, top 1% of t(3) sims

Normal data, same variants (calibration rows)

Verdict per n (rule 2): smallest k with var/mean^2 <= 2/3 and all power diffs within 0.02

• n=25: none (moment-only candidates: k=2, k=3, k=5)
• n=50: none (moment-only candidates: k=3, k=5)

Top-k winsorize sweep: k=2 at n=25 and k=3 at n=50 reach var/mean^2 <= 2/3, but every k loses far more than 0.02 power

Answer to the cycle 84 direction (rule 2 verdict). No k satisfies both halves at either n. Verdict per n, smallest k with capped var/mean^2 at most 2/3 while normal-calibrated power at n=25, 50, 100 stays within 0.02 of uncapped:

• n=25: none. The moment bound alone is met from k=2 (0.53, then 0.35 at k=3 and 0.23 at k=5), but k=1 already loses 0.195 of power at n=25 and 0.097 at n=100, and the loss grows with k.
• n=50: none. The moment bound alone is met from k=3 (0.64, then 0.34 at k=5); k=1 loses 0.180 at n=50 and 0.097 at n=100, k=3 loses 0.41 and 0.31.

Where the residual excess lives (rule 1 share column): after the top k draws are flattened to the (k+1)th largest |draw|, those k draws still carry 0.26 to 0.43 of the AD sum in the top 1% of t(3) sims, against 0.99 for the single largest draw uncapped (cycle 82). The remaining 0.57 to 0.74 sits in the uncapped draws just below the cap: the (k+1)th draw sets the cap level, so heavy t(3) samples still push all k+1 extreme terms out together, and var/mean^2 falls geometrically in k (n=50: 386, 10.3, 1.14, 0.64, 0.34) rather than collapsing at any one k. The excess is not in one draw but in the whole extreme tail, and removing it with a winsorize removes the gate's t(3) signal at the same rate.

Inputs (rule 1)

• Statistic: cycle 71/75 lower-tail-weighted Anderson-Darling gate, sum_i (F_n(i) - Phi(z_i))^2 / Phi(z_i) on standardized data, scaled by n^-1/2; same pure-python code as cycle 82's tail-trim-check.py and cycle 83's cap-single-draw-check.py.
• Script stuff/topk-winsorize-sweep.py, output stuff/topk-winsorize-sweep.csv, runs in about 4 s.
• Seed 84002 (per-stream 8400200 + n + {0 t(3), 1 normal}); 4000 sims per n per law; n = 25, 50, 100; t(3) drawn as gauss / sqrt(gamma(1.5, 2/3)) as in cycle 71.
• Cap rule: variant k replaces the k largest |draw| by the (k+1)th largest |draw|, sign kept, before standardizing (winsorize quantile 1 - k/n, shown in the table). k = 0 is the uncapped baseline row.
• Share: sum of the post-cap AD terms of those k draws divided by the AD sum, averaged over the top 1% of t(3) sims by statistic value.
• Power: critical value is the normal-data p95 of the same variant at the same n (per-n calibration, cycles 76 onward), power is the fraction of t(3) samples above it. "Normal-data power" is read as in cycle 83: the t(3) power of the gate calibrated on normal data (uncapped 0.378, 0.583, 0.835 here against 0.372, 0.585, 0.834 in cycle 83 and 0.36, 0.60, 0.84 in cycle 75).

Rule 1 and rule 2 table: t(3), k = 0 is the uncapped baseline

Power at every n (t(3), gate calibrated on normal-data p95 per n)

Normal data, same variants (calibration rows; power is 0.05 by construction)

Caveats

• The k=1 power loss (0.195 at n=25) matches cycle 83's winsor row (0.17 at n=25), so this is the same effect, not a seed artefact; 4000 sims give a power standard error of about 0.008.
• The winsorize cap floats with the (k+1)th draw, so on normal data it barely moves the critical value (0.078 to 0.082 at n=25 for k=0 to k=2) while on t(3) it removes most of the signal; a fixed-level cap (cycle 83's share caps) is the variant that kept power, and it did not reach 2/3 either.
• Next natural probe: a share cap on the top k terms jointly rather than a winsorize, since the share column shows the residual mass is in the k+1 extreme draws together.